Case 2. If \(u\ne 0\)  and \(v=0\) , then \(v_n\rightarrow 0\)  in \(L^p(\mathbb {R}^4)\) . Let \(\bar{u}_n=u_n-u\) , \(\bar{u}_n\rightarrow 0\)  in \(L^p(\mathbb {R}^4)\) . By the maximum principle (see [1]}), \(u\)  is a positive solution of (REF ), then \(m^-(a_1,0)\le m^-(\Vert u\Vert _2,0)\) .
By the Brézis-Lieb Lemma [2]}, we deduce that
\(\begin{aligned}&P_{a_1,a_2}(u_n,v_n)\\&=||\nabla u_n||^2_2+\Vert \nabla v_n\Vert ^2_2-\mu _1\Vert u_n\Vert ^{4}_{4}-\mu _2\Vert v_n\Vert ^{4}_{4}-\gamma _p\alpha _1\Vert u_n\Vert ^p_p-2\beta \Vert u_nv_n\Vert ^2_2+o_n(1)\\&=||\nabla \bar{u}_n||^2_2+\Vert \nabla v_n\Vert ^2_2-\mu _1\Vert \bar{u}_n\Vert ^{4}_{4}-\mu _2\Vert v_n\Vert ^{4}_{4}-2\beta \Vert \bar{u}_nv_n\Vert ^2_2+o_n(1).\end{aligned}\) 
